There are A000165(3) = 48 permutation of the cube. Half of them are its rotations, forming the subgroup O (the symmetric group S4), and the other half are their inversions.
The inversion is the permutation that exchanges opposite vertices of the cube. It is not to be confused with the inversion of a permutation.

The Cayley table of Oh repeats the pattern of the Cayley table of S4. If, as in this article, the S4 based notation is used, the result of a concatenation of elements of Oh can be derived from the corresponding concatenation of elements of S4: With a,b,c∈S4{\displaystyle a,b,c\in S_{4}} and a′,b′,c′{\displaystyle a',b',c'} being their respective inversions a∘b=c{\displaystyle a\circ b=c} implies a∘b′=a′∘b=c′{\displaystyle a\circ b'=a'\circ b=c'} and a′∘b′=c{\displaystyle a'\circ b'=c}.

As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath productS2≀S3≃S23⋊S3{\displaystyle S_{2}\wr S_{3}\simeq S_{2}^{3}\rtimes S_{3}},and a natural way to identify its elements is as pairs (m,n){\displaystyle (m,n)} with m∈[0,23){\displaystyle m\in [0,2^{3})} and n∈[0,3!){\displaystyle n\in [0,3!)}.But as it is also the direct productS4×S2{\displaystyle S_{4}\times S_{2}}, one can simply identify the elements of tetrahedral subgrup Td as a∈[0,4!){\displaystyle a\in [0,4!)} and their inversions as a′{\displaystyle a'}.

So e.g. the identity (0,0){\displaystyle (0,0)} is represented as 0{\displaystyle 0} and the inversion (7,0){\displaystyle (7,0)} as 0′{\displaystyle 0'}.(3,1){\displaystyle (3,1)} is represented as 6{\displaystyle 6} and (4,1){\displaystyle (4,1)} as 6′{\displaystyle 6'}.

A rotoreflection is a combination of rotation and reflection.While a rotation leaves its axis and a reflection leaves its plane unchanged, a rotoreflection leaves only the center unchanged.

It can be seen that the reverse colexicographic ranks of two inversions (in the sense of point reflections) add up to 40319 - the rank of the inversion(in the sense of point reflection) itself.
This means that their inversion sets(sets of pairs out of their natural order) are complements. (Beware the two different meanings of inversion here.)

Each of the six cubes in the following collapsible boxes shows one of the basic permutations from the top row of the matrix in the bottom left position.
In the other seven positions are the products of applying the reflections along coordinate axes on these basic permutations.

Two permutations (m,n){\displaystyle (m,n)} and (k,n){\displaystyle (k,n)} are complementary to each other, if m+k=7{\displaystyle m+k=7}.
Complementary permutations sum up to a vector of 7s, and their inversion sets are complements,
so their inversion numbers sum up to 28. (Compare one of the number matrices above.)

The conjugacy classes below are always shown in complementary pairs (like inv2/ref1 or rot2/ref2).
The numbers over the triangles are the inversion numbers of the corresponding permutations. It can be seen that corresponding numbers add up to 28.

neut (1)

inv3 (1)

inv2 (3)

ref1 (3)

rot3 (8)

rotref3 (8)

rot2 (6)

ref2 (6)

rot1 (6)

rotref1 (6)

neut (1 × 1)

inv3 (1 × 1)

0

28

inv2 (3 × 1)

ref1 (3 × 1)

12

20

24

16

8

4

rot3 (4 × 2)

6

14

18

18

rotref3 (4 × 2)

22

14

10

10

rot2 (6 × 1)

26

18

20

12

24

8

ref2 (6 × 1)

2

10

8

16

4

20

rot1 (3 × 2)

rotref1 (3 × 2)

6

12

12

22

16

16

Conjugacy classes of square permutations

neut

inv2

ref1

ref2

rot1

0

6

2

4

1

5

3

Examples for tesseract and penteract

The respective numbers of conjugacy classes for 4 and 5 dimensions are 20 and 36.
The following dictionaries show an example pair and the corresponding permutation from each conjugacy class.
(See here for the whole tesseract group.)

Oh has 98 individual subgroups, which are all shown in the list below. (A Python dictionary of them can be found here.)

They naturally divide in 33 bundles of similar subgroups, whose elements belong to the same conjugacy classes.
In this article these bundles are given naive names based on some of the colors assigned to their elements (like Dih4 green orange).
Each of them has a collapsible box below, containing representations of the individual subgroups.

These belong to 25 bigger bundles, which can be identified with a label in Schoenflies or Coxeter notation (like D2d or [2+,4]).
Each of them has a vertex in the big Hasse diagram below.

Four different kinds of Coxeter notation can be distinguished, based on where they contain plus signs:

The symmetry group of the cuboid C23 appears in two essentially different ways as a subgroup of Oh.
The one where the cuboid is the cube itself is the most intuitive one.
In the other one the cuboid is the original cube rotated by 45° around an axis. The one where it is rotated around the z-axis is shown below.
There are 4 individual subgroups (see above).

The symmetry group of the square appears in four essentially different ways as a subgroup of Oh. (C4v or [4] being the most intuitive among them.)
There are 12 individual Dih4 subgroups (see above). Shown below are the four where the square is seen "from above", i.e. a point on the positive z-axis.

In the white box above the colored boxes of the four subgroups are the permutations of the square. Their 2×2 transformation matrices are the top left submatrices of the four 3×3 matrices in the same column. So the last non-zero entry of the 3×3 matrix determines the permutation in this column. (So each column has only two different permutations.) The pattern of these eight last entries identifies the subgroup. It is shown on the left in the little 4×2 matrix under the example solid.

The symmetry group of the triangle appears in two essentially different ways as a subgroup of Oh, with C3v or [3] being the most intuitive among them.
There are 8 individual subgroups (see above). Shown below are the ones where the triangle is seen from a point on the negative main diagonal of the coordinate system.